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Unit Prefixes, Orders of Magnitude, Unit Conversions, Scientific Notation
Scientific Notation Scientific notation can be defined as a system to articulate small and large numbers in a coherent way. In physics, we often deal with very small quantities and very large quantities. Many people believe it is unnecessarily complex and time wasting to continually write out numerical values that have a large amount of digits. The critical purpose of scientific notation is to provide a method of representing numbers in a visual and comprehensible way rather than in an overwhelming jumble of digits. In essence, instead of writing .0000000000000000000007838, we write 7.838x10-22. At first, this may seem confusing and strange. Don’t worry! After some practice with scientific notation, it will become clear that there are countless benefits to using it. The format is undeniably more concise; it supports a deeper comprehension of an exceptionally large/small value in comparison to other relevant values. As we now understand why scientific notation is important to the world of physics, we need to learn how to use scientific notation appropriately. In order to obtain a comfortable regents-level understanding of this system, it is crucial that we know how to convert numbers into scientific notation format. We will also need to know how to perform basic mathematical procedures (such as adding and multiplying) with numbers in scientific notation and how to express scientific notation on a graphing calculator. 'Formulas and Concepts Pertaining To Scientific Notation Format:' The overarching idea in regard to converting long,awkward numbers to scientific notation is to express decimal points as powers of 10. Thus, scientific notation takes on the form: Mx10n M is often referred to as the mantissa. The mantissa must ALWAYS be greater than or equal to one and less than ten (1≤M<10). For example, the number 73000 is written in scientific notation as 7.3x104. It is not written as 730x102 or .73x105! Also, note that the exponent, represented by the variable n, must ALWAYS be an integer. Remembering these rules is crucial in converting to scientific notation correctly. 'Steps for converting numbers into scientific notation:' # Write down the number as a decimal. (The number you want to convert to scientific notation will likely already be expressed as a decimal, but if it is a fraction or a mixed fraction, it must be changed to a decimal format.) # Find the value of the mantissa. This value can be found by inserting a decimal after the first digit and then, dropping all the zeroes. #Find the value of the exponent. To find this, count the number of places between where the original decimal point was located and where it should be located to make the mantissa a value between 1 and 10. (REMEMBER: if the decimal point moves to the right, the exponent is negative and if the decimal point moves to the left, the exponent is positive) #Now that you have values for M and n, just follow the standard format of scientific notation and you are done! Example: Convert .00000008769 into scientific notation. # This number is already expressed as a decimal, thus it does not have to be changed in any way. # The first digit (reminder: this does not include zero) in this number is eight, so a decimal should be inserted after the eight. Now the number is 00000008.769. Next, all the zeros must be dropped. Thus, we can conclude that the mantissa is 8.769. # The number of places from the original decimal point to the mantissa can be determined by simply counting. We can see that there are eight places between .00000008769 and 00000008.769 by observing that there are eight numbers between the two decimal points. We must also take note that the decimal point moves to the right, so the exponent is negative. Thus the exponent value is -8. # Finally, we have all the information to express .00000008769 in scientific notation. We know that M=8.769 and n=8. So, our final answer is… .00000008769 can be expressed in scientific notation as 8.769x10-8 Once we are comfortable with converting numbers into scientific notation, we can learn the rules of multiplying and adding two numbers, both expressed in scientific notation. 'Steps for Multiplying Two Numbers Expressed in Scientific Notation:' # Multiply the two mantissas together # Add the two exponents together # The new mantissa value and the new exponent value should be expressed in standard scientific notation format. Example: Multiply 9.73x10-7 and 2.019x1015 # The product of the mantissas=9.73x2.019=19.64487 # The sum of the exponents=-7+15=8 # (9.73x10-7)( 2.019x1015)=19.64487x108. Although this value is numerically the correct answer, REMEMBER that the mantissa value must be between 1 and 10, so there is still one more step. The decimal point must be moved one space to the left and in reaction, the exponent value must be increased by one. This makes the final answer 1.964487x109. 'Steps for Adding Two Numbers Expressed in Scientific Notation:' # Two numbers expressed in scientific notation can only be added if they have the same exponent, so the first step is to change necessary components of the problem in order to ensure that the exponent value is constant throughout the problem. # Add the mantissas together. # Use the sum of the mantissas as the M value and the constant exponent as the n value in order to obtain the sum in scientific notation format. Example: Add (4.3x105)+(6.2x108) # Either exponent value can be changed to equal that of the other one. In this problem, let’s convert the exponent eight in 6.2x108 to the exponent five. In order for the exponent to decrease by three units, the decimal point must be moved 3 spaces to the right. Thus, 6.2x108 can be written as 6200x105. # The sum of the mantissas=4.3+6200=6204.3 (This is the M value.) # The sum of 4.3x105 and 6.2x108 is 6204.3x105. In order to make the mantissa between 1 and 10, the decimal point must be moved three spaces to the left and the exponent must increase by three. The final answer in scientific notation is 6.2043x108. Now that all of the basic rules of scientific notation and its applications have been outlined, we can learn how the scientific calculator can help us with scientific notation conversions: 'Using A Graphing Calculator To Express Scientific Notation:' # Type in the value of the mantissa # Press “2nd” (the button on the top left underneath “y=”) # Press the “,” button. After this button is pressed, a small uppercase E should appear on the screen next to the mantissa. # Type in the value of the exponent # Press enter (depending on the value of the number, the calculator will either automatically convert it into a regular number or keep it in its scientific notation format). ORDER OF MAGNITUDES Order of Magnitude is used to make approximate comparisons between numbers. Order of Magnitude is the number of powers of ten in a certain number. To calculate the order of magnitude you generally follow the same rules as Scientific Notation. Order of magnitude is used mostly to make comparisons, and produce general relationships between two values. It is never used to calculate precise values, or unequivocal answers. One Order of magnitude equals a power of ten. CHART The chart displays the relationship between scientific notation and Order of magnitude. Calculating the Order of Magnitude is not a difficult procedure, just know that the order of magnitude is approximately what power of 10 a number is raised to. Usually the order of magnitude is equal to the exponent 10 is raised to for example 3.4 * 103 would have an order magnitude of 3. The exception to the rule that the order of magnitude is equal to the exponent I a value, is when the matissa is equal to or over 5. When the matissa is 5 or larger than 5 you round up a power. This is because you want to calculate the approximate value so when the matissa is over 5 its more like multiplying the power of 10 by another 10. For example 8.9* 104 value is closer to 105 than 104. 105 is equal to 100, 000, 104 is equal to 10,000 and 8.9* 104 is equal to 89,000. 89,000 is closer to 100,000 than 10,000. Therefore 8.9* 104 would have an Order of Magnitude of 5. To calculate the difference in order of magnitudes you just subtract the larger magnitude from the other magnitude. And that number again is like a power of 10. For example a number with a magnitude of 8 , is 3 order of magnitudes larger than a number with an order of magnitude or 5. there for the order of the magnitude is 1000 times larger. A real world example is an aunt is approximately .00005* 10-2 m long and a human is approximately .000057* 105 tall. That means a human is 7 order of magnitudes longer than an ant, or you can say that a human is 10,000,000 times larger. ' PICTURE' Order of magnitude is immensely useful to find general ratios between two values, and how much larger or smaller things are. But in order to make accurate comparisons it is important to know the appropriate notation. You must know the accurate rules of Scientific Notation in order to calculate the correct Order of Magnitude. For example if you want to find the approximate relationship between the number 2.167*106 and 25.4*108 say that 25.4*108 was relatively 2 order of magnitudes larger or 100 times larger than 2.167*106. An accurate relationship would be between 2.167*106 and 2.54*109 , you would say that 2.54*109 was about 3 order of magnitudes larger or 1000 times larger. Always remind yourself when finding the ratio that they both must be in the same notation. CHART Order of magnitude is closely related to logarithms. When calculating a logarithm you are calculating, when 10 is raised to what number, equals the number you are taking the log of. For example 10x= 10,000, so if you logged 10,000, the answer would be 3, so x=3. Because 103= 10,000. So if you take a log of a number this would be the Order of magnitude of a number. The rules for rounding, discussed in relation to the matissa do not apply to the method of using logs to calculate the order of magnitude. Because when you round in terms of the order of magnitude based on the matissa you are just changing the number because you should round it to closest power. Where if you round the exponent you may not be rounding to the closest power. For Example Log of 4,000,000 equals approximately 6.602. meaning 106.602 equals to 4,000,0000, and if written in scientific notation it would be 4.0* 10 6. The matissa is still under 5 there for this would still have an order of magnitude of 6. If you rounded the exponents you would have a wrong order of magnitude, you would have 4.0* 107 which is 40,000,000. When solving for magnitude you must know all the rules of Scientific Notation. You must know the order, the relationships, and steps of scientific notation. Use the relationship between logs, scientific notation, units of measurement, and Order of Magnitude to solve for the Order of Magnitude. And most importantly remember that Order of Magnitude is used for approximations and general relationships, not for precise calculations.